Robert C. Merton and Myron S. Scholes have,
in collaboration with the late Fischer Black, developed a
pioneering formula for the valuation of stock options. Their
methodology has paved the way for economic valuations in many
areas. It has also generated new types of financial instruments
and facilitated more efficient risk management in society.

***

In a modern market economy it is essential
that firms and households are able to select an appropriate level
of risk in their transactions. This takes place on financial
markets which redistribute risks towards those agents who are
willing and able to assume them. Markets for options and other
so-called derivatives are important in the sense that agents who
anticipate future revenues or payments can ensure a profit above
a certain level or insure themselves against a loss above a
certain level. (Due to their design, options allow for hedging
against one-sided risk - options give the right, but not the
obligation, to buy or sell a certain security in the future at a
prespecified price.) A prerequisite for efficient management of
risk, however, is that such instruments are correctly valued, or
priced. A new method to determine the value of derivatives stands
out among the foremost contributions to economic sciences over
the last 25 years.

This year's laureates, Robert Merton
and Myron Scholes, developed this method in close
collaboration with Fischer Black, who died in his mid-fifties in
1995. These three scholars worked on the same problem: option
valuation. In 1973, Black and Scholes published what has come to
be known as the Black-Scholes formula. Thousands of traders and
investors now use this formula every day to value stock options
in markets throughout the world. Robert Merton devised another
method to derive the formula that turned out to have very wide
applicability; he also generalized the formula in many
directions.

Black, Merton and Scholes thus laid the
foundation for the rapid growth of markets for derivatives in the
last ten years. Their method has more general applicability,
however, and has created new areas of research - inside as well
as outside of financial economics. A similar method may be used
to value insurance contracts and guarantees, or the flexibility
of physical investment projects.

The problem
Attempts to value derivatives have a long history. As far back as
1900, the French mathematician Louis Bachelier reported one of
the earliest attempts in his doctoral dissertation, although the
formula he derived was flawed in several ways. Subsequent
researchers handled the movements of stock prices and interest
rates more successfully. But all of these attempts suffered from
the same fundamental shortcoming: risk premia were not dealt with
in a correct way.

The value of an option to buy or sell a
share depends on the uncertain development of the share price to
the date of maturity. It is therefore natural to suppose - as did
earlier researchers - that valuation of an option requires taking
a stance on which risk premium to use, in the same way as one has
to determine which risk premium to use when calculating present
values in the evaluation of a future physical investment project
with uncertain returns. Assigning a risk premium is difficult,
however, in that the correct risk premium depends on the
investor's attitude towards risk. Whereas the attitude towards
risk can be strictly defined in theory, it is hard or impossible
to observe in reality.

The method
Black, Merton and Scholes made a vital contribution by showing
that it is in fact not necessary to use any risk premium when
valuing an option. This does not mean that the risk premium
disappears; instead it is already included in the stock
price.

The idea behind their valuation method can
be illustrated as follows:

Consider a so-called European call option that gives the right to
buy one share in a certain firm at a strike price of $ 50, three
months from now. The value of this option obviously depends not
only on the strike price, but also on today's stock price: the
higher the stock price today, the greater the probability that it
will exceed $ 50 in three months, in which case it pays to
exercise the option. As a simple example, let us assume that if
the stock price goes up by $ 2 today, the option goes up by $ 1.
Assume also that an investor owns a number of shares in the firm
in question and wants to lower the risk of changes in the stock
price. He can actually eliminate that risk completely, by selling
(writing) two options for every share that he owns. Since the
portfolio thus created is risk-free, the capital he has invested
must pay exactly the same return as the risk-free market interest
rate on a three-month treasury bill. If this were not the case,
arbitrage trading would begin to eliminate the possibility of
making a risk-free profit. As the time to maturity approaches,
however, and the stock price changes, the relation between the
option price and the share price also changes. Therefore, to
maintain a risk-free option-stock portfolio, the investor has to
make gradual changes in its composition.

One can use this argument, along with some
technical assumptions, to write down a partial differential
equation. The solution to this equation is precisely the
Black-Scholes' formula. Valuation of other derivative securities
proceeds along similar lines.

The Black-Scholes formula
Black and Scholes' formula for a European call option can be
written as

where the variable d is defined by

According to this formula, the value of the call option C,
is given by the difference between the expected share value - the
first term on the right-hand side - and the expected cost - the
second term - if the option right is exercised at maturity. The
formula says that the option value is higher the higher the share
price today S, the higher the volatility of the share
price (measured by its standard deviation) sigma, the higher the
risk-free interest rate r, the longer the time to maturity
t, the lower the strike price L, and the higher the
probability that the option will be exercised (the probability is
evaluated by the normal distribution function N
).

Other applications
Black, Merton and Scholes' method has become indispensable in the
analysis of many economic problems. Derivative securities
constitute a special case of so-called contingent claims and the
valuation method can often be used for this wider class of
contracts. The value of the stock, preferred shares, loans, and
other debt instruments in a firm depends on the overall value of
the firm in essentially the same way as the value of a stock
option depends on the price of the underlying stock. The
laureates had already observed this in their articles published
in 1973, thereby laying the foundation for a unified theory of
the valuation of corporate liabilities.

A guarantee gives the right, but not the
obligation, to exploit it under certain circumstances. Anyone who
buys or is given a guarantee thus holds a kind of option. The
same is true of an insurance contract. The method developed by
this year's laureates can therefore be used to value guarantees
and insurance contracts. One can thus view insurance companies
and the option market as competitors.

Investment decisions constitute another
application. Many investments in equipment can be designed to
allow more or less flexibility in their utilization. Examples
include the ease with which one can close down and reopen
production (in a mine, for instance, if the metal price is low)
or the ease with which one can switch between different sources
of energy (if, for instance, the relative price of oil and
electricity changes). Flexibility can be viewed as an option. To
choose the best investment, it is therefore essential to value
flexibility in a correct way. The Black-Merton-Scholes'
methodology has made this feasible in many cases.

Banks and investment banks regularly use
the laureates' methodology to value new financial instruments and
to offer instruments tailored to their customers' specific risks.
At the same time such institutions can reduce their own risk
exposure in financial markets.

Other research contributions
Besides their valuation method, Merton and Scholes have made
several significant contributions to financial economics. Merton
has developed a new powerful method for analyzing consumption and
investment decisions over time, and generalized the so-called
CAPM (the valuation model for which William Sharpe was awarded
the Prize in 1990) from a static to a dynamic setting. Scholes
has clarified the impact of dividends on stock market values,
together with Black and Miller (Merton Miller was awarded the
Prize in 1990 for his contributions to corporate finance), and
made empirical contributions, for example concerning estimation
of the so-called beta value (a risk measure in the CAPM).

Robert C. Merton, was born in 1944
in New York, USA. He received his Ph.D. in Economics in 1970 at
MIT, Cambridge, USA. He currently holds the George Fisher Baker
Professorship in Business Administration at Harvard Business
School, Boston, USA.

Myron S. Scholes, was born in 1941.
He received his Ph.D. in 1969 at University of Chicago, USA. He
currently holds the Frank E. Buck Professorship of Finance at the
Graduate School of Business and is Senior Research Fellow at the
Hoover Institution at Stanford University, Stanford,
USA